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Did you try to clear data for google play?
I suggest you contact a staff member:
http://talk.sonymobile.com/t5/Welcome/Community-users/m-p/634967
Have you checked so that notifications is enabled in Google Play under Google Play -> Settings -> Notifications? Also make sure that you haven't restricted background data under Settings -> Data usage -> [menu] and try turning of STAMINA under Settings -> Power management if it's enabled.
If that doesn't help, does other push notifications work? Try sending emails to your gmail for example and see if you get a notification. If so, i recommend that you report this to the Google Play support: https://support.google.com/googleplay/?rd=1#contact=1
All other notifications are working.
I never use STAMINA mode, the phone should be capable of all functions ON at all times. I dont want a 50% potential phone when in my pocket.
The other day though google play showed 2 notifications for updates without me restarting it.
But Sony has this nice feature built in "Freeze and crash forcing reboot once every day". So i usually get the notifications after that reboot once a day.
I can just laugh at Sony theese days. A phone that freeze and reboots due to software. Imagine that would happen when you need to call the emergency number 
The problem's still there.
Tried this (and all others solutions proposed on this topic):
Main Menu > Settings > Apps > All > Playstore > Unistall Updates > Launch The Playstore.
Nothing's changed.
So what? No solution to resolve this annoying problem????
Sony xperia Z1 compact D5503 Android KK 4.4.4 stock ROM